Is this what you expect from expectations?
The daily problem for Feb 27 asks which of two special dice (1,1,3,3,5,5) and (2,2,2,2,5,5) has a better chance of winning if both dice are rolled simultaneously and the higher roll wins (In case of a tie, you roll both dice again.)
Notice that both dice have the same expected value.
Suppose we have two six-sided dice, A and B. You are not told what numbers are on the faces, but you are told that A has a greater chance of winning than B.
Let and be the expected values of the rolls of A and B respectively.
Which of the following is true?
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1 solution
Let A be the die ( 1 , 1 , 1 , 1 , 1 , 1 ) and B be the die ( 0 , 0 , 0 , 0 , 0 , b ) , where b > 1 . Then A has a 5/6 chance to win. But E A = 1 and E B = b / 6 . Choosing different values for b , e.g. b = 2 , b = 6 , b = 7 , b = 3 7 we see that there is no order relation between E A and E B that holds in all cases.